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Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation
Mitigating the influence of the boundary on PDE-based covariance operators
Courant Institute, New York University, 251 Mercer street, New York, NY 10012, USA |
Gaussian random fields over infinite-dimensional Hilbert spaces require the definition of appropriate covariance operators. The use of elliptic PDE operators to construct covariance operators allows to build on fast PDE solvers for manipulations with the resulting covariance and precision operators. However, PDE operators require a choice of boundary conditions, and this choice can have a strong and usually undesired influence on the Gaussian random field. We propose two techniques that allow to ameliorate these boundary effects for large-scale problems. The first approach combines the elliptic PDE operator with a Robin boundary condition, where a varying Robin coefficient is computed from an optimization problem. The second approach normalizes the pointwise variance by rescaling the covariance operator. These approaches can be used individually or can be combined. We study properties of these approaches, and discuss their computational complexity. The performance of our approaches is studied for random fields defined over simple and complex two- and three-dimensional domains.
References:
[1] |
C. Bekas, A. Curioni and I. Fedulova, Low cost high performance uncertainty quantification in Proceedings of the 2nd Workshop on High Performance Computational Finance, WHPCF '09, ACM, New York, NY, USA, 2009, Article No. 8.
doi: 10.1145/1645413.1645421. |
[2] |
C. Bekas, E. Kokiopoulou and Y. Saad,
An estimator for the diagonal of a matrix, Applied Numerical Mathematics, 57 (2007), 1214-1229.
doi: 10.1016/j.apnum.2007.01.003. |
[3] |
J. Besag,
On a system of two-dimensional recurrence equations, Journal of the Royal Statistical Society. Series B (Methodological), 43 (1981), 302-309.
|
[4] |
T. Bui-Thanh, O. Ghattas, J. Martin and G. Stadler,
A computational framework for infinite-dimensional Bayesian inverse problems Part Ⅰ: The linearized case, with application to global seismic inversion, SIAM Journal on Scientific Computing, 35 (2013), A2494-A2523.
doi: 10.1137/12089586X. |
[5] |
D. Calvetti, J. P. Kaipio and E. Somersalo,
Aristotelian prior boundary conditions, International Journal of Mathematics and Computer Science, 1 (2006), 63-81.
|
[6] |
G. Da Prato,
An Introduction to Infinite-dimensional Analysis, Universitext, Springer, 2006.
doi: 10.1007/3-540-29021-4. |
[7] |
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07, Online companion to [ |
[8] |
L. C. Evans,
Partial Differential Equations, 2nd edition, Graduate studies in mathematics, American Mathematical Society, 2010, URL http://books.google.com/books?id=Xnu0o_EJrCQC.
doi: 10.1090/gsm/019. |
[9] |
M. Hairer, Introduction to Stochastic PDEs, Lecture Notes, 2009. |
[10] |
T. Isaac, N. Petra, G. Stadler and O. Ghattas,
Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet, Journal of Computational Physics, 296 (2015), 348-368.
doi: 10.1016/j.jcp.2015.04.047. |
[11] |
S. G. Johnson, Cubature—Adaptive Multi-dimension Integration, http://ab-initio.mit.edu/wiki/index.php/Cubature. |
[12] |
Lin, Lu, Ying, Car and W. E,
Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems, Communications in Mathematical Sciences, 7 (2009), 755-777.
doi: 10.4310/CMS.2009.v7.n3.a12. |
[13] |
F. Lindgren, H. Rue and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 423-498, URL http://dx.doi.org/10.1111/j.1467-9868.2011.00777.x.
doi: 10.1111/j.1467-9868.2011.00777.x. |
[14] |
A. Logg, K.-A. Mardal and G. N. Wells (eds.),
Automated Solution of Differential Equations by the Finite Element Method, vol. 84 of Lecture Notes in Computational Science and Engineering, Springer, 2012.
doi: 10.1007/978-3-642-23099-8. |
[15] |
B. Øksendal,
Stochastic Differential Equations, Springer, 2003.
doi: 10.1007/978-3-642-14394-6. |
[16] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010, Print companion to [ |
[17] |
L. Roininen, J. M. J. Huttunen and S. Lasanen,
Whittle-Matérn priors for Bayesian statistical
inversion with applications in electrical impedance tomography, Inverse Problems Imaging, 8 (2014), 561-586.
doi: 10.3934/ipi.2014.8.561. |
[18] |
H. Rue and S. Martino,
Approximate Bayesian inference for hierarchical Gaussian Markov random field models, Journal of Statistical Planning and Inference, 137 (2007), 3177-3192.
doi: 10.1016/j.jspi.2006.07.016. |
[19] |
D. Simpson, F. Lindgren and H. Rue,
In order to make spatial statistics computationally
feasible, we need to forget about the covariance function, Environmetrics, 23 (2012), 65-74.
doi: 10.1002/env.1137. |
[20] |
D. Simpson, F. Lindgren and H. Rue,
Think continuous: Markovian Gaussian models in spatial statistics, Spatial Statistics, 1 (2012), 16-29.
doi: 10.1016/j.spasta.2012.02.003. |
[21] |
A. Singer, Z. Schuss, A. Osipov and D. Holcman,
Partially reflected diffusion, SIAM Journal on Applied Mathematics, 68 (2008), 844-868.
doi: 10.1137/060663258. |
[22] |
A. M. Stuart,
Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.
doi: 10.1017/S0962492910000061. |
[23] |
J. M. Tang and Y. Saad,
A probing method for computing the diagonal of a matrix inverse, Numerical Linear Algebra with Applications, 19 (2012), 485-501.
doi: 10.1002/nla.779. |
[24] |
S. R. Varadhan,
Probability Theory, Courant Lecture Notes in Mathematics, 7. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/cln/007. |
[25] |
P. Whittle,
Stochastic-processes in several dimensions, Bulletin of the International Statistical Institute, 40 (1963), 974-994.
|
show all references
References:
[1] |
C. Bekas, A. Curioni and I. Fedulova, Low cost high performance uncertainty quantification in Proceedings of the 2nd Workshop on High Performance Computational Finance, WHPCF '09, ACM, New York, NY, USA, 2009, Article No. 8.
doi: 10.1145/1645413.1645421. |
[2] |
C. Bekas, E. Kokiopoulou and Y. Saad,
An estimator for the diagonal of a matrix, Applied Numerical Mathematics, 57 (2007), 1214-1229.
doi: 10.1016/j.apnum.2007.01.003. |
[3] |
J. Besag,
On a system of two-dimensional recurrence equations, Journal of the Royal Statistical Society. Series B (Methodological), 43 (1981), 302-309.
|
[4] |
T. Bui-Thanh, O. Ghattas, J. Martin and G. Stadler,
A computational framework for infinite-dimensional Bayesian inverse problems Part Ⅰ: The linearized case, with application to global seismic inversion, SIAM Journal on Scientific Computing, 35 (2013), A2494-A2523.
doi: 10.1137/12089586X. |
[5] |
D. Calvetti, J. P. Kaipio and E. Somersalo,
Aristotelian prior boundary conditions, International Journal of Mathematics and Computer Science, 1 (2006), 63-81.
|
[6] |
G. Da Prato,
An Introduction to Infinite-dimensional Analysis, Universitext, Springer, 2006.
doi: 10.1007/3-540-29021-4. |
[7] |
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07, Online companion to [ |
[8] |
L. C. Evans,
Partial Differential Equations, 2nd edition, Graduate studies in mathematics, American Mathematical Society, 2010, URL http://books.google.com/books?id=Xnu0o_EJrCQC.
doi: 10.1090/gsm/019. |
[9] |
M. Hairer, Introduction to Stochastic PDEs, Lecture Notes, 2009. |
[10] |
T. Isaac, N. Petra, G. Stadler and O. Ghattas,
Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet, Journal of Computational Physics, 296 (2015), 348-368.
doi: 10.1016/j.jcp.2015.04.047. |
[11] |
S. G. Johnson, Cubature—Adaptive Multi-dimension Integration, http://ab-initio.mit.edu/wiki/index.php/Cubature. |
[12] |
Lin, Lu, Ying, Car and W. E,
Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems, Communications in Mathematical Sciences, 7 (2009), 755-777.
doi: 10.4310/CMS.2009.v7.n3.a12. |
[13] |
F. Lindgren, H. Rue and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 423-498, URL http://dx.doi.org/10.1111/j.1467-9868.2011.00777.x.
doi: 10.1111/j.1467-9868.2011.00777.x. |
[14] |
A. Logg, K.-A. Mardal and G. N. Wells (eds.),
Automated Solution of Differential Equations by the Finite Element Method, vol. 84 of Lecture Notes in Computational Science and Engineering, Springer, 2012.
doi: 10.1007/978-3-642-23099-8. |
[15] |
B. Øksendal,
Stochastic Differential Equations, Springer, 2003.
doi: 10.1007/978-3-642-14394-6. |
[16] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010, Print companion to [ |
[17] |
L. Roininen, J. M. J. Huttunen and S. Lasanen,
Whittle-Matérn priors for Bayesian statistical
inversion with applications in electrical impedance tomography, Inverse Problems Imaging, 8 (2014), 561-586.
doi: 10.3934/ipi.2014.8.561. |
[18] |
H. Rue and S. Martino,
Approximate Bayesian inference for hierarchical Gaussian Markov random field models, Journal of Statistical Planning and Inference, 137 (2007), 3177-3192.
doi: 10.1016/j.jspi.2006.07.016. |
[19] |
D. Simpson, F. Lindgren and H. Rue,
In order to make spatial statistics computationally
feasible, we need to forget about the covariance function, Environmetrics, 23 (2012), 65-74.
doi: 10.1002/env.1137. |
[20] |
D. Simpson, F. Lindgren and H. Rue,
Think continuous: Markovian Gaussian models in spatial statistics, Spatial Statistics, 1 (2012), 16-29.
doi: 10.1016/j.spasta.2012.02.003. |
[21] |
A. Singer, Z. Schuss, A. Osipov and D. Holcman,
Partially reflected diffusion, SIAM Journal on Applied Mathematics, 68 (2008), 844-868.
doi: 10.1137/060663258. |
[22] |
A. M. Stuart,
Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.
doi: 10.1017/S0962492910000061. |
[23] |
J. M. Tang and Y. Saad,
A probing method for computing the diagonal of a matrix inverse, Numerical Linear Algebra with Applications, 19 (2012), 485-501.
doi: 10.1002/nla.779. |
[24] |
S. R. Varadhan,
Probability Theory, Courant Lecture Notes in Mathematics, 7. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/cln/007. |
[25] |
P. Whittle,
Stochastic-processes in several dimensions, Bulletin of the International Statistical Institute, 40 (1963), 974-994.
|






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